Programming with classical quantum datatypes

Transcription

1 Programming with classical quantum datatypes Robin Cockett & Brett Giles University of Calgary FMCS 2006, Programming with classical quantum datatypes p. 1/49

2 Yet another Quantum Programming Language (QPL)... Linear type system with (classically controlled) datatypes No higher-order constructs... Only built-in types are Qbit and Int. Built-in unitary transformations. A QPL program consists of function definitions and main instructions. Compiles to Quantum Stack Machine (QSM). Quantum assembler calculates density matrix. FMCS 2006, Programming with classical quantum datatypes p. 2/49

3 Semantics... Countably infinite bi-products of completely positive matrices with trace less than or equal to 1. fhilb CPM(fHilb) Mat (CPM(fHilb)) Similar to the semantics of Peter Selinger s QPL. FMCS 2006, Programming with classical quantum datatypes p. 3/49

4 This QPL... Can declare (classically controlled) datatypes (e.g. lists of Quantum bits). Linear variables are unscoped: live from introduction to first use (cannot be used twice). Classical data extracted from linear variables by use command. Classical variables are scoped: live to end of a block (multiple uses permitted). Quantum bits can be measured, Integers have usual arithmetic functions... FMCS 2006, Programming with classical quantum datatypes p. 4/49

5 Example datatype declarations type Bool = { True False } type L i s t a = { N i l Cons ( a, ( L i s t a ) ) } type E i t h e r a b = { L e f t ( a ) Right ( b ) } type Maybe a = { Nothing Just ( a ) } type BinTree a = { Leaf ( a ) Br ( ( BinTree a ), ( BinTree a ) ) } FMCS 2006, Programming with classical quantum datatypes p. 5/49

6 Language constructs continued... Function declaration rotate :: (n : Int,x 1 : Qbit,x 2 : List(Qbit); y 1 : Qbit,y 2 : List(Qbit)) = {statements} Function calls: rotate(n,x,y;x,y); (x,y) = rotate(n,x,y); rotate(n) x y; Had q;not q;not q 1 <=q 2 ;... FMCS 2006, Programming with classical quantum datatypes p. 6/49

7 Example Program: coin flip # Import Prelude. qpl c f l i p : : ( ; b : Bool ) = { q = 0 > ; Had q ; measure q of 0 > => { b = False } 1 > => { b = True } ; } main : : ( ) = { b = c f l i p ( ) ; } FMCS 2006, Programming with classical quantum datatypes p. 7/49

8 Language construct: measurement measure q of 0 {...} 1 {...};... measurement is a control construct... FMCS 2006, Programming with classical quantum datatypes p. 8/49

9 Example Program: Teleport prepare : : ( ; a : Qbit, b : Qbit ) = { a = 0 > ; b = 0 > ; Had a ; Not b <= a } / / C o n t r o l l e d Not t e l e p o r t : : ( x : Qbit ; b : Qbit ) = { ( a, b ) = prepare ( ) ; Not a <= x ; Had x ; measure a of 0 > => { } 1 > => { Not b } ; measure x of 0 > => { } 1 > => {RhoZ b } } FMCS 2006, Programming with classical quantum datatypes p. 9/49

10 Language constructs continued... Loops:... handled by recursion. Pattern matching: case q of nil {...} cons(x,xs) {...}; FMCS 2006, Programming with classical quantum datatypes p. 10/49

11 Example Program: append # Import Prelude. qpl append : : ( x : L i s t ( a ), y : L i s t ( a ) ; z : L i s t ( a ) ) = { case x of N i l => { z = y } Cons ( v, vs ) => { z = Cons ( v, append ( vs, y ) ) } } FMCS 2006, Programming with classical quantum datatypes p. 11/49

12 Language constructs continued... Discarding: discard q 1,...,q n ; implicit discarding of variables on joining of control paths (with warning). Linear assignment is basic: q = expression; q is a linear variable... There is syntactic sugar for classical assignment (see below). FMCS 2006, Programming with classical quantum datatypes p. 12/49

13 Using classical data... Syntactic sugar: use x 1,...,x n in {statements} x := expr;...; x = expr; use x in {...;} use x 1,...,x n ;...; use x 1,...,x n in {...;} FMCS 2006, Programming with classical quantum datatypes p. 13/49

14 Example Program: rotate r o t a t e : : ( n : Int, h : Qbit, z : L i s t ( Qbit ) ; h : Qbit, z : L i s t ( Qbit ) ) = { case z of N i l => { discard n ; z = N i l } Cons ( x, y ) => { use n in { R( n ) x h ; m = n+1 } ; r o t a t e (m) h y ; z = Cons ( x, y ) } } FMCS 2006, Programming with classical quantum datatypes p. 14/49

15 Example Program: Quantum Fourier Transform q f t : : ( qf : L i s t ( Qbit ) ; qf : L i s t ( Qbit ) ) = { case qf of N i l => { qf = N i l } Cons ( head, t a i l ) => { { Had head ; r o t a t e ( 2 ) head t a i l ; q f t t a i l ; qf = Cons ( head, t a i l ) } } FMCS 2006, Programming with classical quantum datatypes p. 15/49

16 A Quantum stack machine The machine state consists of the quantum stack, a classical stack, and a dump. The quantum stack is represented as a tree, with three types of nodes (Qbit, Datatype, Int) and values at the leaves (zero probability branches are not represented). Operations are done primarily on the top node or top few nodes. Rotation of the tree allows us to bring a node to the top. The dump is used to hold intermediate data during instruction execution. FMCS 2006, Programming with classical quantum datatypes p. 16/49

17 Qbit Nodes A Qbit is represented by a node with four branches, labeled by 00, 01, 10 and 11 (equivalent to the density matrix notation of a Qbit). Note that physically, we only store those branches with non-zero values at the leaves. FMCS 2006, Programming with classical quantum datatypes p. 17/49

18 Int nodes Classical data is represented by a node with an arbitrary (finite) number of branches labeled by classical values (e.g. Int values): FMCS 2006, Programming with classical quantum datatypes p. 18/49

19 Datatype nodes A node with a branch for each constructor which is labeled by the constructor name and the variables the construction binds. FMCS 2006, Programming with classical quantum datatypes p. 19/49

20 Binding... Variables bound along a branch of a node cannot be rotated above their point of binding... so can not be manipulated directly in the Quantum stack. FMCS 2006, Programming with classical quantum datatypes p. 20/49

21 Example program evaluation: coin flip... # Import Prelude. qpl c f l i p : : ( ; q : Qbit, r : Bool ) = { q = 0 > ; Had q ; measure q of 0 > => { q = 0 > ; r = True } 1 > => { q = 1 > ; r = False } ; } main : : ( ) = { ( q, b ) = c f l i p ( ) ; } FMCS 2006, Programming with classical quantum datatypes p. 21/49

22 Empty stack FMCS 2006, Programming with classical quantum datatypes p. 22/49

23 Introduce Qbit 0 FMCS 2006, Programming with classical quantum datatypes p. 23/49

24 Hadamard FMCS 2006, Programming with classical quantum datatypes p. 24/49

25 Measure FMCS 2006, Programming with classical quantum datatypes p. 25/49

26 Left branch FMCS 2006, Programming with classical quantum datatypes p. 26/49

27 Left branch: introduce Qbit 0 FMCS 2006, Programming with classical quantum datatypes p. 27/49

28 Left branch: introduce True FMCS 2006, Programming with classical quantum datatypes p. 28/49

29 Swap to right branch FMCS 2006, Programming with classical quantum datatypes p. 29/49

30 Right branch: introduce Qbit 1 FMCS 2006, Programming with classical quantum datatypes p. 30/49

31 Right branch: introduce False FMCS 2006, Programming with classical quantum datatypes p. 31/49

32 Merge FMCS 2006, Programming with classical quantum datatypes p. 32/49

33 Quantum Stack Machine Instructions QPL programs compile to Quantum Stack Machine (QSM) code. QPL QSM QSM code is run by the QSM-assembler. The design of an efficient yet basic instruction set for this machine has been an issue... FMCS 2006, Programming with classical quantum datatypes p. 33/49

34 QSM Instructions: node construction QLoad x k : creates a node called x with a single branch labeled k on top of the Quantum stack. QCons x cons: creates a node called x with a single branch labeled cons (the specific constructor) on top of the Quantum stack. QMove x: removes the top value of the classical stack and creates, on top of the Quantum stack, a node called x with a single branch labeled by that classical value. QBind z: expects a node with a single branch and binds the variable z down that branch. FMCS 2006, Programming with classical quantum datatypes p. 34/49

35 Construction Transitions (QLoad x k :C,S,Q,D) = (C,S,x:[ k Q],D) (QCons x c:c,s,q,d) = (C,S,x:[c{} Q],D) (QBind z 0 :C,S,x:[c{z 1,...,z n} Q],D) = (C,S,x:[c{z 0,z 1,...,z n} Q[z 0/z 0 ]],D) (QMove x:c,v:s,q,d) = (C,S,x:[ v Q],D) FMCS 2006, Programming with classical quantum datatypes p. 35/49

36 QSM Instructions: node destruction QUnbind x: Expects the stack have a node with a single branch on top with a list of bound variables. The first bound variable is removed and renamed to x to make it free. QDiscard: Discards the top node by merging its classical branches. FMCS 2006, Programming with classical quantum datatypes p. 36/49

37 Destruction Transitions (QDiscard:C,S,x:[ k Q],D) = (C,S,Q,D) (QDiscard:C,S,x:[c{} Q],D) = (C,S,Q,D) (QDiscard:C,S,x:[ v Q],D) = (C,v:S,Q,D) (QUnbind y:c,s,x:[c{z 1,...,z n} Q],D) = (C,S,x:[c{z 2,...,z n} Q[y/z 1]],D) ( k { 0, 1 }) FMCS 2006, Programming with classical quantum datatypes p. 37/49

38 QSM Instructions: stack manipulation QPullup x: Rotates the node labeled x to the top of the Quantum stack. QApply 0 Had: Expects 1 Qbit on top of the Quantum stack and applies the Hadamard transformation to it. QApply 1 Rotate: Parametrizes the Rotate transform by the top element of the classical stack and then applies it to the 2 Qbits on the top of the stack. QName x y: Renames the (visible) node x to y. FMCS 2006, Programming with classical quantum datatypes p. 38/49

39 Manipulation Transitions (QPullup x:c,s,q,d) = (C,S, pull(x,q),d (QApply n t:c,v 1 : :v n :S,Q,D) = (C,v 1 : :v n :S, transform([v 1,...,v n ],t,q),d) (QName x y:c,s,q,d) = (C,S,Q[y/x],D) FMCS 2006, Programming with classical quantum datatypes p. 39/49

40 QSM Instructions: Quantum control; Use; Split; Measure Use label: Expects top of the Quantum stack to be classical. For each branch put this value on the classical stack, perform the computation at label, which is ended by EndQC, and merge the results. Split [Nil label 1,Cons label 2 ]: Expects a datatype node on top of the Quantum stack. Executes the code at label i, which is ended with EndQC, on each branch with a matching label and merges the results. Meas label 1 label 2 : Expects a Qbit node on top of the Quantum stack. Executes the code at label 1, which is ended with EndQC, on the 0 branch and label 2 on the 1 branch and merges the results. FMCS 2006, Programming with classical quantum datatypes p. 40/49

41 Transitions for Quantum Control - Use (Use C U :C,S,x:[ v i Q i ],D) = (EndQC,S, 0, Qc(S, [(x i :v i Q i, C U )], C, 0):D) (EndQC,S,Q, Qc(S, [(x i :v i Q i, C U )] i=j,...,m, C,Q ):D) = (C U,S,x j, Qc(S, [(x i :v i Q i, C U )] i=j+1,...,m, C,Q + Q ):D) (EndQC,S,Q, Qc(S, [], C,Q ):D) = (C,S,Q + Q,D) FMCS 2006, Programming with classical quantum datatypes p. 41/49

42 Transitions for Quantum Control - Split (Split [(c i, C i )]:C,S,x:[c i {V i } Q i ],D) = (EndQC,S, 0, Qc(S, [(x i :c i {V i } Q i, C i )], C, 0):D) (EndQC,S,Q, Qc(S, [(x i :c i {V i } Q i, C i )] i=j,...,m, C,Q ):D) = (C j,s,x j, Qc(S, [(x i :c i {V i } Q i, C i )] i=j+1,...,m, C,Q + Q ):D) (EndQC,S,Q, Qc(S, [], C,Q ):D) = (C,S,Q + Q,D) FMCS 2006, Programming with classical quantum datatypes p. 42/49

43 Transitions for Quantum Control - Measure (Meas C 0 C 1 :C,S,x:[ 0 Q 0, 1 Q 1, Q],D) = (EndQC,S, 0, Qc(S, [(x k : k Q k, C k )] k {0,1}, C, 0):D) (EndQC,S,Q, Qc(S, [(x k : k Q k, C k )] k {0,1}, C,Q ):D) = (C 0,S,x 0, Qc(S, [(x 1 : 1 Q 1, C 1 )], C,Q + Q ):D) (EndQC,S,Q, Qc(S, [(x 1 : 1 Q 1, C 1 )], C,Q ):D) = (C 1,S,x 1, Qc(S, [], C,Q + Q ):D) (EndQC,S,Q, Qc(S, [], C,Q ):D) = (C,S,Q + Q,D) FMCS 2006, Programming with classical quantum datatypes p. 43/49

44 QSM Instructions: classical control Jump label: Jumps to the code at label. CJump label: Pops the top element of the classical stack if it is true jumps to the code at label. Call label: Pushes the return code pointer onto the dump and jumps to the subroutine code. Return: Pops the code label off the dump and continues execution of the code to which it points. FMCS 2006, Programming with classical quantum datatypes p. 44/49

45 Transitions for Classical Control (Jump C J :C,S,Q,D) = (C J,S,Q,D) (CJump C J :C, False:S,Q,D) = (C J,S,Q,D) (CJump C J :C, True:S,Q,D) = (C,S,Q,D) (Call n C C :C,v 1 : :v n :S,Q,D) = (C C, [v 1,...,v n ],Q,R(S, C):D) (Return n,v 1 : :v n :S,Q,R(S, C):D) = (C, [v 1,...,v n ]:S,Q,D) FMCS 2006, Programming with classical quantum datatypes p. 45/49

46 QSM Instructions: classical CGet n: Puts the n th Classical stack value on top of the Classical stack. CApply ADD: Pops the top values off the classical stack, applies the operation and pushes the result back onto the stack. CLoad n: Pushes the constant value n onto the Classical stack. CPop: Pops the top element off the Classical stack. FMCS 2006, Programming with classical quantum datatypes p. 46/49

47 Transitions for Classical Operations (CPop:C,v:S,Q,D) = (C,S,Q,D) (CGet n:c,v 1 : :v n :S,Q,D) = (c,v n :v 1 : :v n :S,Q,D) (CApply op n :C,v 1 : :v n :S,Q,D) = (C, op n (v 1,...,v n ):S,Q,D) (CLoad n :C,S,Q,D) = (C,n:S,Q,D) FMCS 2006, Programming with classical quantum datatypes p. 47/49

48 Recursion The machine stack is actually a stream of stacks, where stacks further down the stream correspond to progressive unfoldings of the recursion. The limit of the stream is the computation. Instructions are lifted by the standard Kliesli lifting to produce a function between streams of Quantum stacks. FMCS 2006, Programming with classical quantum datatypes p. 48/49

49 Conclusion Where are we... State of QPL compiler: Working, generating code! State of quantum stack machine: Working with this instruction set, revising for controlled transforms. State of QPL programs: order finding for Shor s factoring algorithm compiling, and quantum search not done. Where are we going... Expect to complete implementation this fall. Plan to make web runnable version available. Executable compiler and QPL programs will be down-loadable. FMCS 2006, Programming with classical quantum datatypes p. 49/49